Combined Filtering and Parameter Estimation for Discrete-time Systems Driven by Approximately White Gaussian Noise Disturbances

Abstract

We consider a partially observable, discrete-time process{xt, θt, yt} over a finite horizon T, where the unobservable components are {xt, θt}. Conditionally on {θt}, the pair {xt}, {yt} satisfies a linear model of the form (1) below; {θt} itself evolves according to a given joint a-priori distribution p(θ0,…, θT), The purpose of the paper is to determine recursively the joint conditional distribution p(xt, θt|yt), (yt: = {y0,…,yt}), or, more specifically, E{f(xt, θt)|yt}, namely the (mean squre)optimal filter for a given When θtis constant our problem becomes that of the combined filtering and parameter estimation.The optimal filter is computed for the ideal situation of white Gaussian noises and it is shown that, when this filter is applied to a more realistic situation where the noises are only approximately (in the sense of weak convergence of measures) white Gaussian and also {θt} has only approximately the given distribution p(θ0,…,θT), then it remains almost (mean-square) optimal with respect to all alternative filters that are continuous and bounded functions of the past observations.